DNA double-strand break end synapsis by DNA loop extrusion

DNA double-strand breaks (DSBs) occur every cell cycle and must be efficiently repaired. Non-homologous end joining (NHEJ) is the dominant pathway for DSB repair in G1-phase. The first step of NHEJ is to bring the two DSB ends back into proximity (synapsis). Although synapsis is generally assumed to occur through passive diffusion, we show that passive diffusion is unlikely to produce the synapsis speed observed in cells. Instead, we hypothesize that DNA loop extrusion facilitates synapsis. By combining experimentally constrained simulations and theory, we show that a simple loop extrusion model constrained by previous live-cell imaging data only modestly accelerates synapsis. Instead, an expanded loop extrusion model with targeted loading of loop extruding factors (LEFs), a small portion of long-lived LEFs, and LEF stabilization by boundary elements and DSB ends achieves fast synapsis with near 100% efficiency. We propose that loop extrusion contributes to DSB repair by mediating fast synapsis.


Supplementary Note Overview
To understand how the probability of DSB synapsis is affected by loop extruding factors (LEFs), we developed a probability theory framework for the process and used it to derive an analytical solution for the probability of synapsis. Briefly, we focus on the repair of DNA DSBs by non-homologous end joining (NHEJ). NHEJ repair involves two steps. First, the two DSB ends must be brought into proximity (synapsis). Second, they must be ligated back together. Here we focus on the first step, DSB synapsis. Please note that our goal is not to obtain the most precise analytical expression, but rather to derive a sufficiently accurate expression that we can use to obtain mechanistic intuition for how various loop extrusion mechanisms and parameters affect the efficiency of synapsis. Therefore we make approximations whenever necessary to simplify the mathematical form.
We first consider how the simplest loop extrusion model, which involves just 3 parameters, may facilitate DSB synapsis. After that, we extend this simple model by adding the four individual mechanisms discussed in the main text that improve synapsis efficiency: LEF stabilization by boundary elements (BEs); the presence of a subpopulation of long-lived LEFs; LEF stabilization by the DSB ends; and targeted loading of LEFs at DSB ends. Finally, we derive an analytical expression that combines all four additional mechanisms.
We note that our theory does not consider the effect of passive diffusion. Since passive 3D diffusion likely contributes to synapsis in cells in a manner that is synergistic with loop extrusion, we note that our estimates of DSB synapsis efficiencies should be considered lower bounds. Putative roles of loop extrusion in DSB end synapsis (1) (2) Two possible outcomes:

The probability of synapsis mediated by loop-extruding factors
(1) Loop extrusion successfully mediates DSB end synpasis (2) Loop extruion fails to achieve DSB end synpasis nucleus Gap-bridging LEFs extrude gaps

gap-briddging LEFs
Pathway for successful LEF-mediated DSB synapsis. (1) Loop extrusion may facilitate successful DSB end synapsis in two steps: (i) the constraining LEF prevents the two DSB ends from diffusing apart after DSB and (ii) additional gap-bridging LEFs loaded within the loop extruded by constraining LEF can extrude sub-loops to bring the two DSB ends into proximity (2) If the constraining LEF falls off before the two DSB ends are brought into proximity by gap bridging LEFs, the two DSB ends may diffuse apart. In our simulations, we assume synapsis always fails once no constraining LEF remains for a given DSB.
We describe a general form for the probability of synapsis mediated by LEFs, Psynapsis. For the purpose of understanding the limitations of what LEFs can and cannot do for synapsis, we omit contributions from 3D diffusion in all subsequent calculations of Psynapsis. Therefore, in order for LEF-mediated synapsis to happen, at least one constraining LEF must reside over the DSB site at the time when the DSB occurs. We denote the probability of having at least one constraining LEF as P constrained , which can be calculated as the probability that the DSB occurs inside a DNA loop extruded by a LEF. If the condition of having at least one constraining LEF over the DSB is met, then additional LEFs loaded into the gap between the DSB and the edges of the constraining LEF can extrude loops to bring the DSB ends into proximity to achieve synapsis (path (1) in the figure above). We refer to this process as gap-bridging and the LEFs that mediate gap-bridging as gap-bridging LEFs. We note that at least one constraining LEF needs to remain throughout the gapbridging process until synapsis is simultaneously achieved on both sides. If the constraining LEF dissociates before gap-bridging on both sides is achieved, we assume that the two DSBs ends can diffuse apart, and we consider this a failed LEF-mediated synapsis event (path (2) in the figure above). We denote the conditional probability of finishing gap-bridging given that the DSB ends are constrained at the time of DSB occurrence as P end-joining|constrained .
Thus, the probability of synapsis, Psynapsis, can then be expressed as the product of P constrained and P end-joining|constrained : Psynapsis = P constrained · P end-joining|constrained .
In the following sections, we derive analytical estimates for both P constrained and P end-joining|constrained .

The probability of DSB ends being constrained
As stated above, P constrained is simply the probability that the DSB occurs inside a DNA loop such that the broken DSB ends remain constrained by at least one LEF. Since we assume that DSBs occur homogeneously throughout the genome, P constrained is equivalent to the fraction of the genome that is extruded into loops by LEFs. For the scenario without BEs, the fraction of the genome inside loops can be estimated by considering the nesting of LEFs [1]: where l is the average DNA loop size, and d is the LEF separation, i.e., the average linear distance between LEF loading sites. This expression accounts for the increasing chance of LEFs loaded into existing loops and thus not contributing to increasing the fraction of genome inside loops.
To incorporate the effect of BEs on P constrained , we account for how BEs decrease the amount of DNA that is inside loops by prematurely stalling LEFs. Therefore, the fraction of genome extruded into loops becomes: where P unextruded,BEstalling is the probability of DNA becoming unextruded (i.e. unlooped) due to stalling of LEFs at BEs. To calculate P unextruded,BEstalling , we adapted the mean-field theoretical model previously derived by Banigan and Mirny [2] for the fraction of genome coverage by loops. Therefore, where G loop,noBEs is the average length of genome inside loops with no BEs present, and thus if the total genome length is G, G loop,noBEs = G · P constrained,noBEs . The fraction of G loop,noBEs that remains looped in the presence of BEs is given by 1 − P unextruded,BEstalling . Np is the total number of parent loops as defined in [2] (i.e. it is the total number of loops minus the number that exist within a larger loop).
The unknown quantities now become f unextruded,i and l unextruded,i , which we can calculate by enumerating the types of configurations which lead unextruded DNA because of a BE. Instead of explicitly enumerating and calculating all the possible configurations, we take a mean field approach and consider the leading three configurations as being representative of the much larger space of configurations. We first consider the case where the loading sites of two adjacent LEFs are separated by a BE (i.e. are in adjacent topologically associated domains, (TADs)). We refer to this case as Configuration I, as shown in the figure below.
In our mean field approach, this scenario arises if the average distance between the BE and the loading site of LEFs (i.e. d 2 ) is smaller than the TAD size, D, i.e. d 2 ≤ D. In this case, unextruded DNA arises if and only if one LEF is stalled by BEs and the other LEF fails to reach BE. The probability of a LEF motor subunit being stalled by BEs can be estimated by the fraction, [BE-LEF] [LEFo] , where [BE-LEF] is the concentration of stalled LEF-BE complexes and [LEFo] is the total density of LEF motor subunits extruding in one direction: Thus the overall fraction of LEFs in Configuration I can be calculated as the following: where M is the random variable representing the extrusion distance of one LEF motor subunit before the LEF unloads. The pre-factor of 2 accounts for the fact that the LEF stalled by BE could be either on the left or on the right of the BE. If we assume M is exponentially distributed, since each LEF motor subunit (a LEF is composed of o BE stalling can generate three types of LEF pair configurations leading to unextruded DNA. (a) When d 2 ≤ D, the loading sites of the LEF pair are in adjacent TADs, separated by a distance of d. On average, the BE is equidistant to both LEF loading sites. Unextruded DNA arises if and only and if one LEF is stalled by BE and the other LEF fails to reach BE, leading to an unextruded segment of average length calculated in Eq.(10).(b) When d 2 > D, the loading sites of the LEF pair are separate by a TAD. Unextruded DNA can arise with two different LEF pair configurations. The first configuration is identical to the configuration in (A), which occurs when one LEF is stalled by BE, and the other LEF is not stalled by the additional BE in between but fails to reach the leftmost BE. The second configuration occurs when both LEFs are stalled by BEs, resulting in an unextruded segment with the same length as the TAD size.
Next we consider the cases where the loading sites of the adjacent LEFs are separated by a TAD (i.e. two BEs). This occurs when d 2 > D. There are altogether two such types of configurations which we call Configuration II and Configuration III. Configuration II, occurs when one LEF is stalled by one of the BEs forming a BE-LEF complex, whereas the other LEF is not stalled, but fails to reach the BE-LEF complex thereby leaving an unextruded gap (left branch in panel (b) of the figure above). Let [LEF] be the concentration of extruding LEFs not bound to BE. Configuration II type of scenarios occur with the following frequencies and result in the following lengths of unextruded DNA: Configuration III arises when both LEFs are each stalled by one of the BEs, leading to an unextruded DNA segment of average length that equals the TAD size D (right branch in panel (b) of the figure above). Note that while in reality there could be more than one TAD is in between the loading sites of two adjacent LEFs, we do not consider those scenarios within our parameter space under mean-field theoretical model, since the largest separation d we consider is 500 kb, and the smallest total size of two adjacent TADs in our simulations is 600 kb (200 kb and 400 kb TAD next to each other), and thus on average there would not be more than one TAD between the loading sites of two adjacent LEFs. Configuration III type of scenarios occur with the following frequencies and result in the following lengths of unextruded DNA: l unextruded,Configuration III = D.
Combining Eqs.(2)-(3),Eqs.(6)- (15), we obtain the expression for P constrained : [LEFo] [LEF] The average DNA loop size l has been previously estimated using processivity λ and separation d with ∼ 1% precision for λ d ϵ[10 −1.5 , 10 5.5 ], by applying the following expression obtained through fitting a 7-th degree polynomial to simulation results [1]: where: Now the only unknown in Eq. (16)  , we consider the following system: where BE-LEF is the complex of BE and LEF formed upon a LEF being stalled by a BE, with rate constant kassociation. LEF in complex with BE dissociates from the DNA and reloads on the genome with rate constant k dissociation . We assume one BE can only associate with one LEF at a time. Although we consider two-sided LEFs, the extrusion along both directions are identical and independent processes, thus we can consider the model in Eq. (20) only for extrusion in one direction, without loss of generality.
Since the rate of BE-LEF formation is determined by the rate of collision between BE and LEF, the association rate constant kassociation can be written as the following: where v is the unobstructed extrusion speed in one direction (i.e., 1/2 the total extrusion speed), and b is boundary strength, defined as the probability of BE stalling LEF extrusion upon encountering (with probability 1 − b, LEF extrudes past BE). Once a LEF is stalled by a BE, the average duration before the LEF dissociates from the DNA is given by λ 2v , where λ is the processivity, i.e., the average length of DNA extruded by an unobstructed LEF. Thus the dissociation rate constant k dissociation can be written as the following: We can write the following differential equation to describe the rate of formation of BE-LEF: At steady state, we have: Combining Eq.(21)-Eq.(24), we get: Now we can substitute Eqs.(25)- (26) into Eq. (16) to solve for P constrained . In the simulations we have four different TAD sizes of Djϵ{200 kb, 400 kb, 800 kb, 1200 kb}, each of which appears with a frequency of ωjϵ{0.5, 0.25, 0.125, 0.125} respectively, and thus P constrained can be computed by summing Eq. (16) with different TAD sizes weighted by the frequency of the TAD size (assuming b > 0): The theoretical expression we derived for P constrained in Eq. (27) can predict the percentage of DSB sites constrained by LEFs with reasonably high accuracy, as shown in the figure below. For the rest of the paper, we use the weighted average TAD size ofD = 4 j=1 Dj · ωj = 450 kb to simplify equations and calculations, unless specified otherwise. Theory prediction of the fraction of DSB sites constrained by LEFs is consistent with simulation results across different boundary strengths. The middle panel with boundary strength = 0.5 is the same as main text Fig. 3b. P constrained can be predicted given processivity, separation, and boundary strength. Heatmaps of predicted (left) and simulated (right; numbers in brackets show standard error of mean, n=3 independent 1D simulations, with 218 DSB events per simulation) fraction of constrained DSB sites with different combinations of processivity (y axis) and separation (x axis). Each row corresponds to boundary strengths of 0.3, 0.5 and 1 respectively.

Simplified expression for the probability of DSB ends being constrained
By using a linear approximation (tangent line approximation) of [BE-LEF] [LEFo] , we can simplify the expression for P constrained . To this end, we define: Then the linear approximation of h(x), H(x), can be written as: (29), we obtain the linear approximation of Eq. (25): Substituting Eq. (26) and Eq.(30) into Eq. (16), we get the simplified expression for P constrained : Eq.(31) shows P constrained is a monotonic increasing function of l and λ, and a monotonic decreasing function of d.

The probability of end joining given that DSB ends are constrained
To calculate the probability of gap-bridging given that DSB ends are constrained, P end-joining|constrained , we want to compute how often simultaneous gap-bridging on both sides of the DSB happens before the constraining LEF unloads. In other words, we want to determine how frequently the time it takes to achieve synapsis is shorter than the lifetime of the constraining LEF. Since gap-bridging on both sides of the DSB are independent of each other, it is more mathematically tractable to consider the probability of gap-bridging on each side, and then take into account that synapsis requires the gap-bridging on both sides of the DSB to happen at the same time. Therefore, we need to formulate the time to bridge the gap and the lifetime of constraining LEFs respectively, which we will discuss next.

The gap-bridging time distribution
We define the gap-bridging time, T , as the duration between DSB occurrence and the first time that the gap is bridged on one side of the DSB. Thus, let the first-passage time, T , be a random variable and the probability distribution of T be given by fT . To simplify the theory, we initially assume there is no gap-bridging LEFs present between the constraining LEF and DSB ends at the time of DSB occurrence (we modify this assumption later on considering the full probability of gap-bridging), and we assume only one gap-bridging LEF carries out the bridging from beginning to end. Therefore, we can conceptualize gap-bridging on one side of the DSB for a given DSB site as a two-step process: first, a gap-bridging LEF must load between the DSB end and the constraining LEF; second, the gap-bridging LEF must extrude to bridge the gap between the DSB end and the constraining LEF. We define the loading time random variable as X, corresponding to the time it takes to finish the first step, and the extrusion time random variable as Y , corresponding to the time it takes to complete the second step. Thus,

The loading time distribution
Generally, we can assume that the process of loading gap-bridging LEFs into the gap between DSB end is Markovian, and thus exponential, with parameter, k load = ⟨τ load ⟩ −1 , where k load is the LEF loading rate, and ⟨τ load ⟩ is the average loading time. We assume LEFs reload randomly (uniform loading probability across the genome) as soon as they unload. Let, L, be the length of the gap, i.e., the length of DNA between the DSB end and the edge of the constraining LEF. We assume the LEF lifetime is exponentially distributed with an average lifetime of λ 2v . The probability density function(PDF) of the loading time X, defined on the interval xϵ[0, ∞), can be described by the following exponential function: in which: where, as a reminder, v is the extrusion speed in one direction (i.e., 1/2 the total extrusion rate), λ is the processivity (i.e., the average length of DNA extruded by an unobstructed LEF), L is the length of the DNA between the DSB end and the edge of the constraining LEF, and d is the average linear distance between LEF loading sites. As can be told from Eq.(34), the longer the gap, L, between the DSB and the edge of the constraining LEF, the faster the loading of a gap-bridging LEF.

The extrusion time distribution
Provided that a gap-bridging LEF has been loaded in the gap, we can then determine the distribution of extrusion times. Since we assume that only one gap-bridging LEF extrudes the gap, the extrusion time is determined by the time it takes to extrude to whichever of the DSB and constraining LEF is furthest away: is the loading point of the LEF, as shown in the diagram below. Thus, the minimum extrusion time of L 2v , is achieved when the gap-bridging LEF loads right in the center of the gap, whereas the maximum extrusion time of L v is achieved when the gap-bridging LEF loads at either boundary of the gap. Since we assume the loading of the LEFs is spatially homogeneous, the extrusion time Thus, the PDF of the extrusion time Y is given by the following uniform distribution: defined on the interval [ L 2v , L v ]. We can rewrite Eq.(35) as the following to simplify expressions later on: where, Θ = 1 if the expression in the parentheses is True 0 otherwise (37)

Computing the gap-bridging time distribution through convolution
Having determined the loading time distribution, fX , and the extrusion time distribution, fY , we can now compute T = X + Y . To do so we need to compute the convolution of fX and fY : The solution has two parts, making it piece-wise continuous: which results in:

The constraining LEF lifetime distribution
Since gap-bridging must take place before the constraining LEF unloads, we need to also compute the distribution of the constraining LEF lifetime. Let the lifetime of constraining LEFs be given by the random variable C. The probability distribution of C is given by fC . As above, we assume the lifetime of constraining LEFs are exponentially distributed, and ⟨τc⟩ is the average lifetime of constraining LEFs. The PDF of the constraining LEF lifetime C can be expressed as: in which:

The probability of gap-bridging on one side of the DSB
Having determined the gap-bridging time distribution fT , and the constraining LEF lifetime distribution fC , we can now compute the probability of gap-bridging on the side of the DSB that is bridged first. The calculation of fT above assumes there is no gap-bridging LEF present at the time of DSB occurrence. However, there is a nonneglibible probability that gap-bridging LEFs are already present in the gap. The average number of gap-bridging LEFs within the constraining LEF, n, can be approximated as the following using Eq.(17) [1]: where a is defined in Eq.(18)- (19). Let the number of gap-bridging LEFs inside the constraining LEF, N , be a random variable. Then N follows Poisson distribution: Thus the probability of having no gap-bridging LEFs inside the constraining LEF is: Conversely, the probability of having one or more gap-bridging LEFs is: As an approximation, we assume if one or more gap-bridging LEFs are present within the constraining LEF at the time of DSB occurrence, then one gap will be successfully bridged with probability 1. This approximation is motivated by the observation that the loading of gap-bridging LEF is the rate-limiting step of synapsis (Supplementary Fig.  3). If one or more gap-bridging LEFs are present within the constraining LEF, one of the two gaps are likely closed by the pre-existing gap-bridging LEFs. However, if initially there is no gap-bridging LEF inside the constraining LEF, the gap-bridging time T must be smaller than the constraining LEF lifetime C. We note fT is a function of the gap length L, and we denote the gap length on the side bridged first as L 1 st . Therefore, where FC is the cumulative distribution function of the constraining LEF lifetime: Because fT is piece-wise continuous, we can compute this by breaking it up into two parts:

Simultaneous gap-bridging on both sides of the DSB and the gap-bridging LEF lifetime distribution
In order to achieve synapsis, the gaps on both sides of the DSB must be bridged simultaneously. In other words, once the gap on one side of the DSB is bridged, the gap on the other side of the DSB must be bridged before the gap-bridging LEF on the side bridged first unloads. Let the lifetime of gap-bridging LEFs that have already finished gap-bridging on the side of DSB bridged first be given by the random variable G, whose PDF is fG. Then the probability of gap bridging on the second side (while the gap-bridging LEF on the first side remains bound) can be written as: where: Utilizing the memoryless property of exponential distribution, the PDF of the lifetime G of the gap-bridging LEF that have already finished gap-bridging on the side bridged first, can be expressed as: in which: Note the PDF of gap-bridging LEF lifetimes is mathematically identical to the PDF of constraining LEF lifetimes, since these two kinds of LEFs are essentially identical, and their identity are assigned based on their locations relative to a DSB. We assign different notations here to facilitate our discussions of extensions to the loop extrusion model later where distinguishing them becomes helpful. Now we can compute the probability of bridging the gap on the other side of the DSB while the gap-bridging LEF on the side bridged first remains:

Computing the probability of joining DSB ends given DSB ends are constrained
Since gap-bridging on the first side and the second side of the DSB are independent processes, the probability of both gaps being bridged upon the first try is the product of Eq.(52) and Eq.(59). However, gap-bridging does not need to be achieved upon the first try: as long as the constraining LEF remains, even if the gap-bridging LEF on the side bridged first unloads, the gap-bridging process can continue until simultaneous gap-bridging on both sides of the DSB is fulfilled. Thus the probability of joining DSB ends given DSB ends are constrained for a given DSB site whose gap lengths are L 1 st and L 2 nd can be written as the following infinite sum: Utilizing the formula for the infinite sum of a geometric series, the equation above can be reduced to: Eq.(61) is written for a specific DSB site with gap lengths of L 1 st and L 2 nd for the side bridged first and the side bridged second respectively. We assume the total length of the gaps on both sides of the DSB (the sum of L 1 st and L 2 nd ), is the average DNA loop size l: We can then integrate over L 1 st ϵ(0, l) to generalize Eq.(61) for any DSB site, given that the location of DSB is random: We define: Thus Eq.(63) can be rewritten as: We have computed Eq.(64) and Eq.(65) for specific gap lengths of L 1 st and L 2 nd respectively in Eq.(52) and Eq.(59). We rewrite them below for the general gap length of L: Notice Eq.(66) is simply the difference between A(L) and B1(L): Now we can substitute Eqs.(17)- (19) and Eqs.(68)-(70) into Eq.(67) and perform numerical integration to obtain P end-joining|constrained .
The probability of synapsis Psynapsis can then be determined by multiplying the numerically integrated P end-joining|constrained and the P constrained computed in Eq. (27). Note that P end-joining|constrained and P constrained both only depend on the LEF processivity λ and the LEF separation d, and thus Psynapsis can also be determined as long as we know λ and d. Below we compare the synapsis efficiency predicted by our analytical expression for Psynapsis and the simulated synapsis efficiency with different combinations of λ and d. This comparison shows that our theory is reasonably accurate: Theory prediction of synapsis efficiency is consistent with simulation results. Heatmaps of predicted (left) and simulated (right; numbers in brackets show standard error of mean, n=3 independent 1D simulations, with 218 DSB events per simulation) synapsis efficiency with different combinations of processivity (y axis) and separation (x axis). Boundary strength = 0.5 was used in the simulations.

Simplified expression for the probability of end-joining given the DSB is constrained and two important relative timescales
Our theory above highlights that Psynapsis can be determined from just λ and d. However, despite this seemingly simple picture and the reasonable agreement between our theory and simulations, our theoretical formulation for Psynapsis is still too complicated to be written in a compact form. Therefore, it is challenging for someone to gain mechanistic intuition for what factors underlie efficient synapsis. In this section, we aim to simplify the expression for Psynapsis with the goal of gaining mechanistic intuition. Let us consider the scenario where gaps are bridged upon the first try. Further, let us consider a DSB site with the break right in the center of the constraining LEF, and thus with gap length L = l 2 on both sides of the DSB. We assume there is no gap-bridging LEF present in the constraining LEF at the time of DSB occurrence. Then P end-joining|constrained can be approximated as the following expression simplified from the product of Eq.(52) and Eq.(59): We can rewrite Eq.(71) as: where: We can further simplify Eq.(72) by defining: Then Eq.(72) can be written as: The simplified formula for P gap-bridged|constrained suggests that given the DSB ends are constrained, synapsis efficiency is dictated by two relative timescales: the ratio of loading time and the constraining LEF lifetime and the ratio of extrusion time and the constraining LEF lifetime. Further, P gap-bridged|constrained monotonically decreases with the two relative timescales, so decreasing either or both of the two relative timescales will improve synapsis efficiency.

Synapsis with additional mechanisms
In this section, we will discuss four experimentally plausible extensions of the simple loop extrusion model that improve synapsis efficiency: stabilization of LEFs by BEs, the presence of a small fraction of long-lived LEFs, stabilization of LEFs by DSB ends, and targeted loading of LEFs to DSBs. We start by investigating how the addition of each of the four mechanisms modifies the expressions derived above. Finally, we end by showing the combined effect of all four mechanisms on synapsis efficiency.

Synapsis with stabilized LEFs by BEs
One mechanism that improves synapsis efficiency by increasing P constrained is the stabilization of LEFs by BEs. We define, w, as the fold increase in LEF lifetime through stabilization by BEs. We assume LEFs with one or both motors bound to a BE have identical fold increase in lifetime, w. Intuitively, P constrained increases with BE stabilization of LEFs since the LEFs stabilized by BEs extrude larger loops; therefore, DSBs are more likely to happen inside loops.
To incorporate the effect of BE stabilization on P constrained , we modify the rate of BE-LEF dissociation in Eq.(22) to the following: The updated BE-LEF dissociation rate in Eq.(79) in turn updates the fraction of LEF motor subunits extruding in one direction that are bound to BEs in Eqs.(25)- (26) to the following: To account for the effect of BE stabilization of LEFs on parameters such as the average LEF processivity and average LEF loop size, we first need to know what fraction of LEFs are stalled at BEs. Let, β, be the fraction of LEFs that have at least one motor subunit bound to BEs, then β can be calculated as: Eq.(82) can reasonably well predict the fraction of LEFs stabilized by BEs: The percentage of LEFs stabilized by BEs predicted by our theory is largely consistent with the simulation results. The error bars represent the standard error of mean, n = 3 independent 1D simulations, with 218 DSB events per simulation.
We can then use Eq.(82) to compute the weighted average processivity and the calculate the average loop size: Eqs.(80)-(81) and Eq.(86) in turn update the probability of DSB happening in loops defined in Eq.(16) to the following: We next aim to compute P end-joining|constrained,BEstabilized . The first consideration is to account for the effect of BE stabilization on the distribution of constraining LEF lifetimes, fC . We thus sought to calculate the probability that a DSB is flanked by constraining LEFs. As a first-pass, we calculate the probability that a DSB occurs in a TAD held together by at least one constraining LEF. Since a TAD is defined by a pair of convergent BEs, the probability,P stabilized , of a TAD having at least one BE occupied, is: As shown in the figure above, most of TADs have at least one LEF stabilized by BE in the parameter space considered. Since the LEFs stabilized by BE will likely constrain DSB ends in a DSB-containing TAD due to their prolonged lifetime, we make the simplifying assumption that with LEF stabilization by BE, all constraining LEFs obtain a w fold increase in lifetime.
Thus, the modified constraining LEF lifetime distribution is: where: As a further simplifying approximation, we neglect the effect of BE stabilization on loading time distribution, since only the fraction of LEFs stabilized by BEs have slower dynamics.
The modified fC updates Eqs.(67)-(70) to the following: P synapsis,BEstabilized can then be computed as: Eq.(95) can accurately predict the synapsis efficiency with BE stabilization, validating our mechanistic explanation of BE stabilization facilitating synapsis by improving the chance of DSB happening inside a loop and increasing the lifetime of constraining LEFs:

Synapsis with a small fraction of long-lived LEFs
Another mechanism that increases both the chance of DSB happening in loops and τ constrained is the presence of a small fraction of long-lived LEFs. Let, αo, be the fraction of long-lived LEFs. Let, s, be the fold increase in long-lived LEFs' lifetime compared with normal LEFs. To incorporate the effect of the presence of a small fraction of long-lived LEFs on P constrained , we use an weighted average processivity to update the average loop size and the fraction of LEF motors extruding in one direction that are bound to BEs: l long-lived = 10 a long-lived · d (99) Eqs.(99)-(101) in turn update the probability of a DSB happening in loops defined in Eq.(16) to the following: [LEFo] long-lived Long-lived LEFs are over-represented in the population of constraining LEFs as they extrude larger loops. We therefore approximate the fraction of constraining LEFs that are long-lived LEFs, α, as the fraction of DNA extruded by long-lived LEFs among the DNA extruded by all LEFs assuming all LEFs are unobstructed: Then PDF of the constraining lifetime will be modified to the following: where: ⟨τ c,long-lived ⟩ = sλ 2v As we only consider a small fraction of long-lived LEFs, we neglect its effect on the loading time distribution as a simplifying approximation.
The modified fC updates Eqs.(67)-(70) to the following: P synapsis,long-lived can then be computed as:

Synapsis with LEFs stabilized by DSB ends
Mechanisms that stabilize gap-bridging LEFs will increase the likelihood of simultaneous gap-bridging events on both sides of the DSB. One such mechanism is the stabilization of LEFs by DSB ends which affects the lifetime, G, of the subpopulation the gap-bridging LEFs that have come into contact with the DSB end. Let, r, be the fold increase in G. We note that G was introduced in Section 1.3.4, to account for the lifetime of gap-bridging LEFs that have already finished gap bridging -therefore, as a first order approximation to simplify our calculations, r only affects gap-bridging LEFs that have fully bridge L 1 st , but does not help promote the initial bridging process. The modified PDF of the gap-bridging LEF lifetime can be written as: in which: The modified fG updates Eq.(67) and Eqs.(69)-(70) to the following: Note that since DSB stabilization is a reactive mechanism that only acts post DSB occurrence, it does not change the probability of DSB occurring in loops. Thus P synapsis,DSBstabilized can then be computed as:

Synapsis with targeted loading of LEFs to DSB ends
One mechanism that can reduce τ loading and thereby increase P end-joining|constrained is targeted loading of LEFs to the DSB site. Let, F , be the targeted loading factor (i.e., the fold increase in loading probability at the DSB compared with anywhere else in the genome). Let, U , be the average distance between two adjacent DSBs in kb, and in all simulations of the paper, we use U = 10000 kb, which corresponds to one DSB occurring every 10 Mb. The PDF of the loading time X can then be modified as the following: where: The modified ⟨τ load ⟩ updates Eqs.(67)-(70) to the following: Like DSB stabilization, targeted loading is also a reactive mechanism that only acts post DSB occurrence, it does not change the probability of DSB occurring in loops. Thus P synapsis,targeted can then be computed as:

Synapsis with all four mechanisms combined
Now that we have explored how individual mechanisms facilitate synapsis, we can determine their combined effects. Only BE stabilization and the presence of long-lived LEFs affect P constrained , as both DSB stabilization and targeted loading are reactive mechanisms. To determine the combined effect of BE stabilization and the presence of longlived on P constrained without further complicating the mathematical form of the solutions and to maximally utilize the framework developed above, we first replace λ in Eq.(80) with the weighted average λ long-lived defined in Eq.(96) to obtain an updated expression of [BE-LEF] [LEFo] and [LEF] [LEFo] : Then the fraction of LEFs stabilized by BEs, β, can be updated accordingly: Now we can use the following weighted average processivity to calculate the average loop size, as an approximation of the combined effect of BE stabilization and long-lived LEFs on P constrained : Eqs.(124)-(125) and Eq.(130) in turn update the probability of DSB happening in loops defined in Eq.(16) to the following: For simplicity, we neglect the effects of BE and DSB stabilization, and long-lived LEFs on the loading time as an approximation: We approximated the combined effect of long-lived LEFs and BE stabilization on the constraining LEF lifetime distribution as the following: in which: As an approximation, we neglect the scenario of long-lived LEFs acting as gap-bridging LEFs, since long-lived LEFs are much less likely to be loaded at a DSB to function as gap-bridging LEFs given their slow dynamics. Thus the PDF of the gap-bridging LEF lifetime remains the same as Eq. (111): Subsequently, with all four mechanisms combined, Eqs.(67)-(70) are modified to the following: P synapsis,combined can then be computed as: Eq.(142) can predict the synapsis efficiency with all four additional mechanisms combined with reasonable accuracy, supporting our mechanistic explanation of how these mechanisms come together to facilitate DSB end synapsis:

Simplified expression for the probability of synapsis with four additional mechanisms
We can derive the simplified expression for the probability of synapsis with all four mechanisms combined.
For P constrain,combined , we can first apply the same linear approximation (tangent line approximation) of [BE-LEF] [LEFo] used to obtain Eq. (31): (143) For P end-joining|constrained,combined , similar to above, we can simplify by considering the scenario where gaps are bridged upon the first try (before the first gap-bridging LEF falls off), with gap length L 1 st and L 2 nd on two sides of the DSB. We assume no gap-bridging LEFs are present within the constraining LEF at the time of DSB occurrence. Then P end-joining|constrained,combined can be approximated as the following: where: Now we can write the simplified expression for P synapsis, combined :

Two important relative timescales underpinning synapsis efficiency
Since estimates based on experimental evidence suggest that most of the interphase DNA is inside loops at any given time ( [1,3,4]), P constrained,combined is likely close to 1. Thus achieving close-to-perfect synapsis efficiency hinges on P end-joining|constrained,combined . We next ask what factors underlying P end-joining|constrained,combined have the most dominant impact on synapsis efficiency. Notice the recurrent terms, τ loading τ constrained and τ extrusion τ constrained , with different prefactors in Eq.(144), we define the following weighted relative timescales: On top of our theoretical framework, we also used 1D simulations to determine whether all four mechanisms could combine to improve synapsis efficiency. We added processivity/separation as an additional dimension, resulting in a 5-dimensional parameter scan with 768 different parameter combinations (see main text Fig. 6a). The two weighted relative timescales in Eqs.(147)-(148) can effectively separate the 5-dimensional parameter scan simulation data points based on synapsis efficiency, suggesting lowering the two relative timescales is key to improving synapsis efficiency: Synapsis efficiency (%) The two relative timescales can effectively cluster simulation data points according to their synapsis efficiency. Same as main text Fig. 6b. The color of each data point shows the average synapsis efficiency of n = 3 independent 1D simulations for a given parameter combination, with 218 DSB events per simulation. The two relative timescales for each data point are calculated using Eqs.(147)-(148) based on the input parameters.

Limitations of our analytical theory and model
Despite the close agreement between our theory prediction of synapsis efficiency and the simulation results in the parameter space bounded by experimental estimates (see Supplementary Note 2), it is worth pointing out several important assumptions and approximations made to simplify the mathematical form of the analytical solution: 1. We assume the prolonged LEF lifetime due to additional mechanisms (BE and DSB stabilization, and longlived LEFs) does not affect the loading time Y . While the accuracy of predictions is largely unaffected by this assumption when λ > d as shown above, the assumption no longer holds when λ ≤ d as the prolonged lifetime for a fraction of LEFs reduces the pool of dynamic LEFs that can be quickly loaded at DSB, leading to overestimation of synapsis efficiency: 2. We assume that only one gap-bridging LEF is bridging the gap on each side of the DSB for the calculation of the first-passage time T . This assumption no longer holds when λ ≫ d and leads to underestimation of synapsis efficiency (top right corner of the heatmaps in main text Fig. 3b).
3. We assume if one or more gap-bridging LEFs are present within the constraining LEF at the time of DSB occurrence, one of the two gaps is bridged already. While a seemingly crude approximation to account for the preexisting gap-bridging LEFs' impact on synapsis, this assumption does not significantly affect the accuracy of theory prediction.
Perhaps the most significant limitation of our theory is that we neglect the effect of passive 3D diffusion. While we estimate the time it takes for 3D diffusion alone to bring two DSB ends back into proximity is likely too long to be consistent with the synapsis timescale, 3D diffusion likely acts concurrently and synergistically with loop extrusion to further improve synapsis efficiency. When the two DSB ends are brought close enough by loop extrusion, the chance of the two DSB ends randomly encountering each other through passive diffusion also greatly increases. Therefore, the synapsis efficiency predicted by our theory likely delineates the lower bound of the physiological synapsis efficiency.
Finally, we have also made several biological assumptions that abstract the complex synapsis process into a simplified picture more tractable for theoretical analysis: 1. We assume uniform loading probability of LEFs across the genome aside from DSB ends. While it remains unclear to what extent cohesins show preferential loading to certain genomic regions, previous work suggests that cohesin loading may not be uniform across the genome [5][6][7].
2. We assume uniform LEF extrusion speed throughout the genome and that only BEs may stall LEF extrusion, while it has been shown that other non-BE elements such as RNA polymerases could also act as partial extrusion barriers [5,8,9].
3. We assume a uniform probability of DSB formation throughout the genome, while evidence suggests certain sites such as binding sites for CTCF and cohesin are more prone to generate DSBs [10][11][12][13][14]. Since DSB sites close to CTCF likely benefit more from BE stabilization to improve synapsis efficiency, and DSB sites close to cohesin binding sites are more likely to be constrained by LEFs and have gap-bridging LEFs loaded in the gaps to improve synapsis efficiency, our theory might underestimate the physiological synapsis efficiency by assuming a uniform probability of DSB formation.

Conclusion
In conclusion, we built a probabilistic theoretical model with a relatively simple mathematical form, that can predict the simulated synapsis efficiency in the parameter regime with λ > d with a fairly good accuracy. Our theory highlights two important roles of loop extrusion in synapsis: LEFs can constrain DSB ends and thereby prevent them from diffusing apart, and gap-bridging LEFs can mediate synapsis by bringing the DSB ends back together. Since DSB ends can only be joined by loop extrusion if the DSB occurs inside a loop, loop coverage of the genome is crucial to achieve high synapsis efficiency. Mechanisms that stabilize LEFs before the DSB occurs such as BE stabilization and the presence of a small fraction of long-lived LEFs could serve to improve the coverage of the genome by LEFs. Provided that the DSB ends are constrained, our theory then points to two relative timescales, τ loading τ constrained and τ extrusion τ constrained that underlie synapsis efficiency. By extending our simple theory with 4 physiologically plausible mechanisms, we show how each of the four mechanisms decreases the two relative timescales and thereby improves synapsis efficiency. The final expressions with all four mechanisms combined illustrate the synergistic effects of the mechanisms. Despite the various approximations made in deriving the theory, the relatively good agreement between our analytical theory and simulations lends credibility to our mechanistic insights obtained from our theoretical model.
Our theory can also serve to guide experimental perturbation of the DSB synapsis machinery for further validation the role of different extended mechanisms discussed above, and to explain observations of the perturbations' impact on synapsis efficiency. Our theory has direct implications for the dependence of DSB repair efficiency on genomic context. For example, our theory points to the importance of BE stabilization in efficient synapsis, which predicts reduced synapsis efficiency in genomic region that lacks BEs (CTCF binding sites etc.). Indeed, heterochromatin, which usually has lower density of bound CTCF binding sites, has been shown to be more sensitive to radiationinduced chromosomal aberrations than euchromatin in Chinese hamster cells [15], consistent with our theory prediction.

Supplementary Note Overview
Given the large possible parameter space for individual variables considered in our simulations and modeling as well as the vast parameter combinations generated by permutation, we sought to use prior experimental estimates to generate plausible upper and lower bounds on individual parameter values. In this Supplementary Note, we detail the calculations used to justify the parameter bounds chosen in our study.

Estimation of the range of LEF separation and processivity
Two studies performed absolute quantification of CTCF and cohesin in HeLa cells [4] and mouse embryonic stem cells (mESCs) [3]. Since in simulations we assume LEFs re-load somewhere on the genome immediately after unloading, we could use the density of chromatin-bound cohesin to estimate the LEF separation, d (the inverse of LEF density).
Using fluorescence-correlation spectroscopy (FCS) and fluorescence recovery after photobleaching (FRAP), the chromatin-bound SCC1(subunit of cohesin) copy number has been estimated to be ∼ 160, 000 for HeLa cells [4]. Considering the total HeLa genome length of 7.9 Gb [4], we can estimate the cohesin spearation as the following, assuming cohesin exists as monomeric ring [16]: d Holzmann,monomer = (7.9 · 10 9 bp)/160000 If we assume cohesin extrudes as dimers, then [16]: Through a combination of FCS, "in-gel" fluorescence, and flow cytometry, we previously carried out an absolute quantification of CTCF and cohesin in mESCs. We estimated the cohesin density to be 5.3 per Mb (assuming monomeric) or 2.7 per Mb (assuming dimeric) [3,16], which correspond to the following cohesin separation: Thus far, absolute quantification of cohesin has only been performed in two mammalian cell types, HeLa and mESC. As can be seen, the density varies substantially between these two cell types, and may vary even more among other cell types. It is therefore associated with significant uncertainty. Nevertheless, these two studies bound the LEF separations in the following range: Let, τo, be the LEF lifetime (residence time). Let vT , be the total unobstructed LEF extrusion speed. Then for the two-sided LEFs considered in our study, vT = 2v, where v is the unobstructed extrusion speed in one direction. Then LEF processivity can be computed as the product of total extrusion speed (in both directions) and the LEF lifetime: Several single-molecule experiments for condensins and cohesins [17][18][19][20][21] measured the unobstructed total extrusion speed to be in the range of vT ϵ[0.5 kb/s, 2 kb/s]. Note that the extrusion speed estimated from in vivo measurement is often slower [16,22,23], likely due to various protein roadblocks bound to DNA including BEs like CTCF. While synapsis time calculated from our simulations is inversely proportional to extrusion speed, synapsis efficiency is independent of extrusion speed. Unless otherwise specified, we use vT of 1 kb/s for the calculation of synapsis time.
Multiple studies have estimated cohesin's residence time which varies with cell cycle phase. Focusing on G1, we estimated cohesin's residence time to be ∼22 minutes in mESCs [24], whereas Holzmann et al. estimated it to be ∼13.7 minutes in HeLa cells [4]. In contrast, the residence time of condensin I was determined by FRAP to be ∼2 minutes [25]. Thus the bounds of vT and τo estimated from these studies provide the following range of LEF processivity given Eq.

Estimation of the fold increase in LEF lifetime upon stabilization by BEs
CTCF can increase cohesin's chromatin residence time by interacting with cohesin in such a way that it outcompetes cohesin interactions with WAPL, which unloads cohesin from DNA [26]. An independent study found that CTCF could also facilitate cohesin acetylation and prolong the lifetime of acetylated cohesin [27]. Cohesin contains one of the two variant STAG subunits, STAG1 or STAG2 [28]. CTCF stabilizes both cohesin-STAG1 and cohesin-STAG2, but with a lesser extent for cohesin-STAG2 [27]. Cohesin-STAG1's stable residence time, τ cohesin-STAG1,stable , and dynamic residence time, τ cohesin-STAG1,dynamic , during G1 phase have been determined by inverse fluorescence recovery after photobleaching (iFRAP) [27]: Since the longer lifetime of cohesin-STAG1 could be attributed to both stabilization by CTCF and acetylation of SMC3 (a subunit of cohesin) by ESCO1 and that cohesin-STAG2 is stabilized by CTCF to a lesser extent than cohesin-STAG1 [27], the ratio of the stable residence time of cohesin-STAG1 and the dynamic residence time of cohesin-STAG1 could serve as an upper bound for the fold increase in LEF lifetime due to stabilization by BEs, w: wϵ [1,20] (158)

Estimation of the fraction and lifetime of long-lived LEFs
Long-lived LEFs can be conceptualized as a subpopulations of LEFs with intrinsic longer lifetime due to chemical modifications such as acetylation of the cohesin subunit SMC3. The STAG subunit of cohesin can be composed of either STAG1 or STAG2 given rise to two distinct forms of cohesin. Cohesin-STAG1 is preferentially acetylated during G1 phase, and cohesin-STAG2 contains four times less acetylated SMC3 relative to cohesin-STAG1 [27]. About ∼ 30 − 50% of cohesin-STAG1 is stably bound to chromatin in G1 [24,27,29]. The relative levels of STAG1 and STAG2 varies substantially between cell types [30]. Cohesin-STAG1 constitutes about about 25% of total cohesin in HeLa cell [4], about 33% of total cohesins in immortalized mouse embryonic fibroblasts (iMEFs) [31], and about 55% in normal human bronchial epithelial cells (NHBE) [32], with the rest being cohesin-STAG2. Taken together, if we assume all the stably bound cohesin-STAG1 is acetylated, up to ∼30% of all cohesin is then acetylated, which we use as an upper bound for the fraction of long-lived LEFs, αo, since factors other than chemical modifications such as stabilization by CTCF could also contribute to the stably bound cohesin's longer lifetime [27]: Stably bound SCC1 exhibits about 50 fold increase in lifetime compared with the dynamic fraction of SCC1 [27]. Thus we can bound the fold increase in long-lived LEFs' lifetime compared with normal LEFs: sϵ [1,50] (160) To limit the number of free variables, we use an intermediate value s = 20 throughout our study.

Estimation of the fold increase in LEF lifetime upon stabilization by DSB ends
The ATM kinase at DSB ends is hypothesized to phosphorylate cohesin, thereby increasing cohesin's lifetime [33]. When cohesin is stabilized by ATM kinase at DSB ends, the higher processivity means that there is higher probability that cohesin can extrude all the way to adjacent BEs. About 1.5 fold enrichment of SCC1, a subunit of cohesin, at BEs was observed in DSB-containing TADs relative to SCC1 count prior to DSB occurrence [33]. We performed simulations with different fold stabilization of LEF at DSB ends, and found the ∼1. Simulated LEF enrichment at BEs in DSB-containing TADs is consistent with experimental observations. Fold enrichment is calculated as the number of LEFs in the DSB-containing TADs within 5 kb to the BEs at the indicated time points normalized against the number of LEFs in the same regions prior to DSB occurrence. The error bars represent standard error of mean, n = 3 independent 1D simulations, with the fold enrichment from each simulation overlaid as individual dots on the bar plot. The pink dashed line represents the SCC1 fold enrichment at BEs in the DSB-containing TAD determined by Arnould et al. [33].
Given the uncertainty associated with the experimental measurement, we use 8 as an upper bound for the fold increase in LEF lifetime due to stabilization by DSB ends, r: rϵ [1,8] (161)

Estimation of the fold increase in LEF loading probability at DSB
Enrichment of cohesins at DSBs has been reported in several studies [33][34][35][36], and this enrichment was recently found to be dependent on cohesin loader NIPBL, as well as ATM and MRN complex recruited to DSB sites [33], pointing to a reactive mechanism that targets cohesin to DSB sites. We performed simulations with different fold increase in LEF loading probability at DSB, and found the ∼1.57 fold enrichment of SCC1 in the chromosome 20 DSB-containing TAD of Dlva cells [33] corresponds to about 250 fold increase in LEF loading probability at DSB (Supplementary Fig. 6a). We implemented targeted loading by increasing the loading rate of LEFs within 1 kb of DSB for simplicity, whereas the observed accumulation of LEFs is not limited to the immediate proximity to DSBs but the whole DSB-containing TADs [33], suggesting LEFs might be targeted to larger regions around DSB instead of just the DSB ends. Therefore, we compared the fold enrichment of LEFs in DSB-containing TADs here instead of just comparing the fold enrichment immediately around DSB ends. Given the noise in ChIP-seq experiments and the 2.5-5 times higher fold enrichment of SCC1 within 4kb around DSB reported by Cheblal et al. [36], we use 5000 (corresponding to 7 fold enrichment of LEFs in DSB-containing TADs) as an upper bound for the fold increase in LEF loading probability at DSB, F :        [33] shown in the left. The contact maps are simulated with one parameter combination (indicated in legend) producing ≥ 95% synapsis efficiency. The white arrows highlight the stripe pattern. The stripe pattern grows quickly over time and reaches a steady state by 1 hr after DSB occurrence. Quantitative differences in contact frequency between our 1D simulations and the experiments may be due to differences in 1D simulation implementation versus experimental conditions (see Methods). The pink text labels on top of each simulated Hi-C map indicate % experimental stripe patter size over time.(b) The bar plot shows the synapsis efficiency before and after knocking out two of the five mechanisms discussed in Fig. 6. The error bars represent standard error of mean, n = 3 different parameter combinations that achieved ≥ 95% synapsis efficiency before knock-out, with the synapsis efficiency from each parameter combination overlaid as individual dots on the bar plot.  Fig. 5c. The error bars in all bar plots represent 95% confidence interval of the mean using maximum likelihood estimation of the exponential distribution accounting for censored data [37].